Lattices from codes over $$\mathbb {Z}_q$$ Z q : generalization of constructions D, $$D'$$ D

On Lattice Constructions D and D' from q-ary Linear Codes

“…For this, we first need to establish connections between Constructions D and D' and, subsequently, with Construction A. More details about these connections can be seen in [68,67].…”

“…In the following, the results and definitions cited are adapted versions of [68] using the scaled version of Construction D presented next.…”

“…These constructions are extensively studied, especially for the binary case, and appear in papers such as [56,63,7,75] and references therein. Recent works deal with generalizations of Constructions D, D' and D to linear codes over finite fields [23], codes over the ring Z q of integers modulo q [68,67] and for cyclic codes over finite fields (Construction D (cyc) ) [29]. It is well-known that some remarkable lattices with higher coding gain can be described via binary code Constructions D and D', as turbo lattices [59], the Barnes-Wall lattices [15] and LDPC lattices [57].…”

Bounds for the $$l_1$$ l 1 -distance of q-ary lattices obtained via Constructions D, D $$^{'}$$ ′ and $$\overline{\text{ D }}$$ D ¯

“…In [18] it is shown that the sets Λ D and Λ D respectively obtained via Constructions D and D from Z n q ⊇ C 1 ⊇ C 2 ⊇ · · · ⊇ C a are always lattices of full rank in R n . When a = 1, we have…”

“…We consider here q ∈ N, the ring Z q of integers modulo q and linear codes C ⊆ Z n q (additive subgroups of Z n q ). There are several constructions involving linear codes and lattices, for example the Construction A [3] and the Constructions D, D and D [18]. These constructions appear in recent works such as [1], [4], [5], [6], [7], [12], [13], [14] and [20].…”